Heat the oven to 350°. Coat a rimmed baking sheet with butter, cover it with parchment paper, then coat the paper with butter.
Make the cake. In a small bowl, whisk 2/3 cup of the sugar and the zest together. Add the dry ingredients and whisk until well blended. In a large bowl with a hand mixer set on high, beat together the 5 egg yolks, oil, milk, and vanilla. Combine the mixtures and beat until blended.
In a large bowl with a hand mixer set on high, beat the 4 egg whites until soft peaks form, about 1 1/2 minutes. Add the remaining sugar and continue beating until stiff peaks form, about 1 minute more. Fold the egg whites into the batter, then pour it onto the prepared baking sheet. Use a spatula to spread it evenly. Bake until the top of the cake is light brown, about 15 minutes. Let cool.
Meanwhile, prepare the filling. In a small bowl, toss the berries, sugar, and zest.
In a large bowl with a hand mixer set on high, beat the remaining ingredients until stiff peaks form, about 2 minutes.
Spread the filling onto the cake, leaving a 1-inch border. Scatter the berries on top. Beginning at a short end, roll the cake, letting the parchment fall away as you work. Place the cake seam side down on a platter and dust with confectioners' sugar.